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After having read this post on MathOverflow about the etymology of various terms that are used in abstract algebra like groups, rings and fields, I was wondering how and why did the term algebra come to denote specifically algebra over a field or a commutative ring? Why pick the same name as the discipline to denote a very specific algebraic structure?

This question might not be very useful mathematically, but I'd nevertheless like to know the origin of this nomenclature.


marked as duplicate by Simon S, Mnifldz, Community Aug 10 '15 at 16:03

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    $\begingroup$ With the introduction of History of Science & Mathematics Stack Exchange I've often wondered if questions like this should end up there. $\endgroup$ – user83387 Aug 10 '15 at 14:38
  • $\begingroup$ @shuckles I did not know about the existence of History of Science and Mathematics SE. Thanks for the info..:) $\endgroup$ – sayantankhan Aug 10 '15 at 16:03

A. A. Albert introduced a $K$-algebra $A$ in the sense that $A$ is a $K$-vector space with distributive bilinear product $A\times A\rightarrow A$, $(a,b)\mapsto a\cdot b$. This includes Lie algebras, Jordan algebras, associative algebras, commutative algebras etc. A natural generalization is to replace the field $K$ by a commutative ring $R$ with $1$. For example, one can consider Lie algebras over the ring $\mathbb{Z}$ (although the name Lie ring is used, too).
The same applies to groups, say $GL_n(K)$ over a field $K$. It can be generalized to $GL_n(R)$ over a commutative ring $R$, and still is called a group ("why is a group over a ring called a group - and is not a group ring").


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